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School Science This report addresses both the current state and future potential of laboratory experiences in US at the high school curriculum. It provides clear design principles for the development of laboratory experiences. - NRC, 2003 , BIO2010: Transforming Undergraduate Education for Future Research Biologists This new volume provides a blueprint for bringing undergraduate biology education up to the speed |
of today s research fast track. It includes recommendations for teaching the next generation of life science investigators. - NSF, 1996 , Shaping the Future: New Expectations for Undergraduate Education in Science, Mathematics, Engineering, and Technology (NSF 96-139) This report provides guidelines for more effective use of the investments made by individuals, organizations, and agencies seeking to improve undergraduate education |
in science, technology, engineering, and mathematics (STEM). - NSTA Teacher Resources: National Science Education Standards This PDF document from the National Science Teachers Association outlines NSTA's resources for educators at several levels who want to implement the National Science Education Standards in their classes. This is the 2003 version of standards. - Seymour and Hewitt, 1997 , Talking About Leaving: |
Why Undergraduates Leave the Sciences This books examines the reasons why undergraduate students switch from science, mathematics, and engineering majors to nonscience majors. - Strategic Programs for Innovations in Undergraduate Physics (SPIN-UP): Full Report PDF - Strategic Programs for Innovations in Undergraduate Physics at Two-Year Colleges (SPIN-UP/TYC (more info) ) (1.88 Mb) - Tobias, 1992 , Revitalizing Undergraduate Science: Why |
Some Things Work and Most Don't Every wave of mathematics and science education reform obliterates the one before and leaves little lasting change in its wake. Sheila Tobias' research suggests that the emphasis on curriculum and pedagogy and the seeking after some "magic bullet" are doomed to fail; that innovators, working alone without adequate "buy-in" from their colleagues do not |
improve the quality of instruction overall; and that funders misconstrue the true nature of the problem and of the solution. - Tomorrow's Professor #222: The Urgency of Reinventing Undergraduate Education at Research Universities This is an excerpt from a speech given by Nancy Cantor at the dedication of The Reinvention Center at SUNY - Stony Brook. - US Dept of |
Ed, 2000 , Before It's Too Late, A Report to the Nation from the National Commission on Mathematics and Science Teaching for the 21st Century The primary message of this report holds that America's students must improve their performance in mathematics and science if they are to succeed in today's world and if the United States is to stay competitive |
Dodwell: The Obliquity of the Ecliptic THE USE OF THE GNOMON IN ANCIENT AND MEDIAEVAL OBSERVATIONS; AND THE ERRORS OF OBSERVATION WITH THIS INSTRUMENT [note from the Setterfields: we have found illustrations of the instruments mentioned by Dodwell in this |
chapter and have linked them as a group below the entire text so as not to disturb the flow of the text itself. After each set of pictures for each instrument there is a 'back to text' return link if |
you wish to view them as you read instead of after you have finished the chapter. They are linked with the FIRST time each instrument is mentioned.] The most ancient instrument, and the one most frequently used for measuring the |
mid-day altitude of the sun at the summer and winter solstices, chiefly for the purpose of determining the date of the solstices, was the ordinary or plain gnomon, which was a vertical pillar, or column, fixed on a horizontal surface. |
The length of the shadow cast by the sun was measured, and from this the sun’s altitude was calculated. “For making the more considerable observations, both the ancients and moderns have made great use of it (the vertical gnomon), especially |
the former, and many have preferred it to the smaller quadrants, both as more accurate, easier made, and more easily applied.” Hutton, Philosophical and Mathematical Dictionary 1815. Vol. 1. Article, “Gnomon”. Out of 85 observations, from 1100 B.C., to 1650 |
A.D., the whole of the Chinese observations, and the great majority of the others, including Hindu, Greek, Arabian, Persian and mediaeval European observations, were made with the vertical gnomon. In many cases, where other instruments were used, such as by |
Greek astronomers at Alexandria, and later by some of the Arab and mediaeval European observers, they were supplementary to the vertical gnomon, which was also in use. It is, therefore, chiefly to the errors of this instrument that our attention |
is directed. A circumstance, which complicates some of the observations, is that the ancient observers did not correct their observations for refraction, parallax and the sun’s semi-diameter. The latter correction is required because the edge of the shadow, cast by |
the gnomon, corresponds to the upper edge of the sun, and not to its centre. In certain cases, however, instead of the ordinary or plain gnomon casting a shadow, one was used having a metal plate at the top, pierced |
with a hole. Through this the sun cast a circular spot of light on the horizontal surface. The centre of this spot of light was measured, and gave the angle, [which] correctly referred to the sun’s centre. Nevertheless, a correct |
result was obtainable with the shadow-casting gnomon if both summer and winter solstices were observed; because the true double angle of obliquity is given by observation of the sun’s upper edge on both occasions, affected only slightly by very small |
differences of refraction. When only one solstice was observed, however, as in the observation by Pytheas of the summer solstice at Marseilles in 326 B.C., it is necessary to apply the correction for the sun’s semi-diameter to his recorded observation. |
Very often, the ratio of the height of the gnomon to the length of the shadow, reduced, when possible, to whole numbers, or to whole numbers and fractions, was given, in preference to the angle itself. Thus, Strabo, in speaking |
of the latitude of Alexandria, determined when the sun was on the celestial equator, says that: “At Alexandria, the relation of the gnomon to the shadow on the day of the equinox is as 5 to 3.” It is a |
striking coincidence that when the latitude of Alexandria is calculated from these figures, after allowing for refraction, parallax, and correction from the sun’s edge to the centre, we get the exact latitude of the Museum of Alexandria, where the observations |
were made, viz. 31° 11’ 42” N. Pliny (70 A.D.), in his Natural History, Book II, Chapter 72, similarly gives the ratio in whole numbers for Rome, for he says that at Rome, at the time of the equinoxes, the |
shadow is to the gnomon as 8 is to 9. From this ratio, after allowing for refraction, parallax, and correction from the sun’s edge to the centre, we get for the latitude of Rome 41º 52’ 07” N, which is |
only 1’ 26” less than the latitude of the Capitol at Rome. Whole numbers with a fraction, however, are given by Strabo for the observation, made by Pytheas, of the mid-day shadow of the sun at the summer solstice at |
Marseilles about 326 B.C. The ratio of the height of the gnomon to the length of the shadow is given by Strabo as 120 to 41 4/5. Ptolemy gives the ratio as 60 to 20 5/6 or 120 to 41 |
4/6. Concerning this, R.T. Gunther, in his account of Ancient Surveying Instruments (Early Science in Oxford, Vol. 1, p. 330), says: “About 326 B.C., in preparation for a voyage of discovery which ended in the finding of our island of |
Britain, Pytheas sailed from Phocoea to Marseilles. There he erected a large gnomon divided into 120 parts, and fixed its latitude with a result that seems almost incredibly accurate, for applying the sun’s semi-diameter, which he omitted, the latitude he |
obtained differs not more than one minute from the true latitude of Marseilles Observatory”. These and many similar results from Chinese and other sources, by the substantial accuracy of the latitudes determined with the vertical gnomon, give abundant testimony that |
this instrument, in the hands of experienced observers in ancient and mediaeval times, was capable of giving results close to the truth, when certain corrections, omitted by ancient observers, are applied to them. It is of interest here, as showing |
the familiarity of the ancient people with the use of the vertical gnomon, to mention its wide use for domestic and other purposes in ascertaining the time of day. In Becker’s Charicles, illustrating the private life of the ancient Greeks, |
it is stated that the gnomon was, unquestionably, the most ancient means of measuring the time of day. The shadow which it cast was measured in feet. It is seldom mentioned except in reference to the hour of supper or |
of the bath. For the former a shadow ten or twelve feet long was assigned, and for the latter a six-feet shadow is spoken of. It is therefore probable that the gnomon was usually constructed so as to throw a |
shadow about 12 feet long shortly before sunset, for this was the time at which the supper, or chief meal, usually took place. The hour of bathing was that preceding the chief meal. In Charicles, in a description of a |
wedding, the sun had sunk half way from the meridian, while preparations were still being made in the house of the bride’s parents. The mother, later on, gives warning that the evening will soon be approaching, and requests a female |
servant to ascertain the time, “Go, Menodora, and measure the shadow on the sun-dial (gnomon) in the garden.” “We have the clepsydra (water-clock) here,” interposed Chloris: “see how much water there is left in it; it will run off once |
and measurement of the shadow of the gnomon in ordinary civil life, we may be sure that for scientific purposes the length of the shadow was carefully measured by the ancient astronomers. The most ancient observation of the inclination of |
the earth’s axis, or obliquity of the ecliptic, which has been preserved, has been generally attributed to the celebrated prince of China, Chou Kung, brother of Wu Wang, (founder of the Chou Dynasty), who is renowned in China through all |
the succeeding centuries for his wise legislation and interest in astronomical science. It was made at Loyang, in the province of Honan, China, in about 1100 B.C. This observation was made with a vertical gnomon, 8 feet high. A height |
of 8 feet was the standard height of the gnomon in China, and this was fixed by law, on account of the confusion caused by varying heights of gnomons. The gnomon, in different forms, also goes back to very ancient |
times in Egypt, even as far back as the 15th century B.C. In addition, the Egyptian obelisks, and even the Great Pyramids themselves, were, for practical purposes, essentially gnomons on a gigantic scale. The most ancient obelisk still standing in |
Egypt, (68 feet high), is that of Senusert I of the XII Egyptian Dynasty at Heliopolis. This dates back to about 2050 B.C. The Roman Emperor Augustus, at the beginning of the Christian Era, followed the example of the Egyptians |
in using an obelisk 75 feet high, which he had removed from Heliopolis to Rome, “to mark the shadows projected by the sun.” Pliny, who mentions this fact, also says that “The name of obelisks signifies that they are sacred |
to the sun; it is the image of the sun’s rays which the obelisk reproduces.” This obelisk was erected in the Campus Martius at Rome, and was one of those previously erected by Psamtek I of the XXVI Egyptian Dynasty |
at Heliopolis. It is now situated on Monte Citorio, near the present Italian House of Parliament. The shadows of the sun were cast upon a level stone corresponding in size to that of the obelisk. They were measured with a |
brass scale, inserted in the stone, and the length of the days and nights was thus determined, together with the solstices. As the summer solstice approached, the midday shadow decreased little by little, and afterwards increased again; and the opposite |
effects were observed at the winter solstice. Manilius, the mathematician, in order to obtain more exact observations, fixed on the apex of this obelisk a golden ball, which cast a dense black circular shadow on the pavement. The centre of |
this could be easily and accurately measured. This form of gnomon, with a terminal ball, is found depicted on medals of the time of Philip of Macedon (359 B.C.); and this arrangement, which was common at Rome, is believed to |
have been introduced into Greece by Menelaus, King of Sparta, about 1180 B.C. Manilius added these words, concerning his measurements: “This observation now for nearly 30 years is not consistent, either through the discordant course of the sun itself, and |
some change in the sky, or through some change in the universal earth, by which it has moved away from its centre, as I have detected myself, and I hear of also in other places.” This is a remarkable statement, |
and constitutes the first recognition of a progressive change in the obliquity of the ecliptic. It also confirms the accuracy obtainable with the vertical gnomon, in the hands of an ancient astronomical observer. The use of the gnomon also extended |
far into the past among the astronomers of Chaldea; and Herodotus says that it was from the Chaldeans that the early Greek astronomers learnt the use of the gnomon and the polos. The polos was a later development of the |
gnomon. It was like a basin, in the centre of which stood the vertical staff, or gnomon, and on it the twelve hours of the day were marked with lines. The sun’s altitude was also measured with it. The gnomon, |
in some of its forms, was called by the Greeks a “skiotheron” (shadow-taker), or “helioptropion,” i.e., an instrument for indicating the turning of the sun at the solstices. It was furnished with measuring marks, and was used to show “the |
solstices, the times, the seasons, and the equinox.” Not only the Chinese and the Greeks, but the ancient Hindus also used the gnomon for similar purposes. Vitruvius (about 30 B.C.) enumerates more than 10 different forms of the gnomon; and |
while some, used as sun-dials, were portable, others attained considerable dimensions, up to 20 to 30 feet in height. The highest gnomon, if we except the Great Pyramids of Egypt, was that of Ulugh Beigh; it was used at his |
Observatory in Samarkand in 1437 A.D., and was 180 feet high. In addition to the gnomon, circular instruments were used by Greek These were the armilla, the astrolabe, and the quadrant. The armilla and the astrolabe were divided into 360 |
degrees, and the quadrant into 90 degrees. It is said that the larger instruments of this kind were subdivided to every 10 minutes of arc. This would permit of still closer measurement by estimation of fractions of the subdivisions. The |
simplest form of the armilla consisted of a large copper ring, graduated with degrees and subdivisions, and mounted vertically in the meridian. A plumb-bob was used for assuring verticality. Within it a second ring rotated in the same plane. This |
carried two diametrically opposite pins, with pointers for reading the circle. The altitude of the sun at noon was measured by turning the inner ring until the shadow of the upper pin fell centrally on the lower one. It is |
clear that the Greek astronomers at Alexandria did not restrict themselves to any one instrument, but used several that were available for various kinds of observation. It has been said that “their development of geometry and trigonometry, their invention of |
well-designed instruments, and their measurement and re-measurement of celestial objects, enabled them to test their carefully formulated theories of the solar system.” Although circular instruments came into use at this period, enabling altitudes of the heavenly bodies to be read |
off directly from these instruments, without calculation, nevertheless, the very great majority of solar observations of ancient times were made with the vertical gnomon. Ptolemy, known as the “prince of astronomers,” who made astronomical observations at Alexandria from 127 A.D. |
to 151 A.D., invented a notable improvement of the gnomon. This was called Ptolemy’s Organon Parallactikon, or Regula Ptolemaica, afterwards called Ptolemy’s Rules, or Triquetrum. This instrument was a vertical post, carrying two hinged arms. The shorter upper arm was |
hinged at the top of the post. The longer lower arm was hinged at a distance from the top exactly equal to the length of the upper arm. The lower extremity of the upper arm slid along the lower arm, |
and carried a pointer, with which the scale marks in the lower arm read. This arrangement formed an isosceles triangle, the equal sides of which were both 60 divisions in length; the scale divisions of the base enabled the angle |
at the vertex to be obtained very simply from Ptolemy’s “table of chords.” Ptolemy used the apparatus for measuring the zenith distance of the moon at Alexandria. A large foresight to take the entire disc of the moon, and a |
small backsight, were used. Gunther, in describing this instrument, points out that “in after years, with Ptolemy's own instrument, the Triquetrum, Copernicus made those measurements with which he overthrew the Ptolemaic System, and gave us a new idea of the |
Universe.” The Triquetrum used by Copernicus was 8 feet in height. It was made of pine wood, and divided by ink marks, the two equal arms into 1000 parts, and the long rule into 1414 parts. In illustration of the |
accuracy of this instrument, it is to be mentioned that the obliquity of the ecliptic obtained from the observations made by Copernicus, after allowing for modern corrections for refraction etc., is within 23 seconds of arc, and that which is |
similarly obtained from Ptolemy’s observations of the moon, is within 15 seconds of arc of the true value. Although the ancient astronomers measured only the length of the shadow cast by the gnomon, and calculated the altitude of the sun |
without correcting for refraction, parallax, or the sun’s semi-diameter, they nevertheless took great care to ensure the verticality of the gnomon, and the horizontality of the surface on which the shadow fell. They also measure the length of the shadow |
with considerable precision. It is easy to apply the corrections they omitted; and the substantial accuracy of the latitude, which are then derived from their observations, proves that their solar observations were not affected by gross errors, as has been |
sometimes supposed, but are of a higher order of accuracy than has been generally admitted. Let us now consider the means which they employed to obtain the best results from their standard instrument, the gnomon. In the first place, verticality |
was effected by the use of the plumb-line, which was a well known accessory used in ancient astronomical observations. Dr. Breasted describes the Egyptian “Merkhet,” used for orienting temples from times earlier than 2000 B.C., and onwards. It consisted of |
a sight vane made from the middle rib of a palm leaf, and a plumb-line suspended from an ivory support. It is also known that in the stellar records of the tombs belonging to the XX Dynasty in Egypt, an |
observer is shown making an astronomical observation with an instrument furnished with a plumb-line. The plumb-line, kept taut with a small leaden plummet or plumb-bob, thus goes back to the earliest times. In the description of Ptolemy’s Astrolabe it is |
stated that “when an observation was made, the instrument was kept vertical by a plumb-line.” In China the most ancient astronomical classic book, the Chou Pei, parts of which go back to 1100 B.C., does not mention the use of |
a plumb-line, but instructions are given for setting up the gnomon (Pei) in a circle (Chou), in which, after leveling the surface on which the shadow falls, a set-square is used to establish verticality of the gnomon. The set-square had |
its sides in the ratio of 3, 4, 5, or better 6, 8, 10, in which 8 feet is the height of the gnomon, 6 feet the base of the right angled triangle, and 10 feet its hypotenuse. The setting |
up of the gnomon, and its use in observation, were superintended by an officer called the Ta-Tsiang (Grand Carpenter). The French astronomer, E. Biot, who translated the Chou Pei into a French rendering in 1841, comments, He also points out |
that the Chou Pei “refers in express terms to the gnomon with a hole, (which produces a circular image of the sun, in place of the shadow edge cast by the ordinary or plain gnomon). This, until now, has passed |
as having been introduced into China by the Arabs about the 13th century of our era.” Thus we see that, as in the case of the magnetic compass, the ancient Chinese led the way many centuries before the European advance |
in science took place. The introduction of the gnomon with a hole, according to the indications of the text of the Chou Pei, and the date of the commentators, is said to be several centuries after the time of Chou |
Kung, and possibly as early as the 4th or 5th century B.C., though the flat topped gnomon was more usually employed. The Arab Astronomers in the Middle Ages gave the following rules for assuring the verticality of the gnomons which |
they used: (1) In order to obtain horizontal lines and surfaces, the ancient astronomers made use of water in receptacles of various dimensions. Thus: “It was in pouring water into large cavities of small depth, that the Egyptians succeeded in |
giving a perfect horizontality to the base of their pyramids.” (2) It is stated that when the ancient observers prepared a level surface for setting up a gnomon, “the surface of the ground was flattened and leveled until water poured |
on it ran off equally on every side.”(3) A similar method was used by the ancient Hindu astronomers, as described in the Surya Siddhanta. The ancient Hindu Observatory..... “consisted principally of a leveled horizontal plane, a floor or terrace of |
Chunam, which is a lime made from shells, and which, when dry, is hard and capable of receiving a polish equal to that of marble. On the surface of this chunam floor, leveled with water, a circle is described, and |
a vertical rod of given length is erected at the centre, as a style or gnomon, and by means of the length and direction of its shadow cast on the plane by the sun, a variety of astronomical problems are |
solved." (4) Vitruvius (B.C. 63 to A.D. 14) described both “libra aquaria” (water levels), and “chorobates.” The latter was the simplest kind, consisting of “a long wooden trough or canal; which being equally filled with water, its surface shows the |
line of level.” (5) In China a gnomon 8 feet high is described by Du Halde, at Pekin, which cast a shadow on a table, having a brass plate in the middle, on which was traced a meridian line 15 |
feet long. “All round the table are small channels to receive the water, whereby it is to be leveled.”(6) This was evidently a device in keeping with ancient practice in China. Among the Arabs it is mentioned in the Encyclopedia |
of Islam (Houtsma, Vol. III, page 535, article Al-Mizan), that the Arab astronomers certainly adopted a large number of methods of leveling, and testing levels, from other peoples, either the Byzantines or the Persians. Statements about the making of canals |
etc. agree with those of Vitruvius, who in turn drew on Greek sources. The Arabs learnt partly from Greek works, but they also utilized data gained from practical experience. To examine if the surface was perfectly horizontal, the following tests |
were adopted: Errors of the Ancient Observations We can now enter into the question of the errors of observation associated with these ancient instruments and methods by which the obliquity of ecliptic was found. This is of great importance in |
the present study, because large errors, if they existed in the ancient observations, would be the only valid astronomical ground of objection to the conclusions derived from them. If the ancient observations were so erroneous as to be completely unreliable, |
then we should be following a false clue, leading us away from the truth. If, on the other hand, these old observations are, in general, reliable within comparatively small limits of error, then the truth which they establish may well |
be of the highest importance, and may lead us into new vistas concerning the mystery of the life of man upon the earth. Any other astronomical objection, such as a suggestion that some kind of curve, other than a curve |
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